Adding outlines to hair particles?
For silhouette outline is usually used Freestyle, but it is not working with Hairs. Since Freestyle is post process effect anyway, there is even probably faster to use Compositor to calculate this effect.

You can try some node tree based on object separation - like Index OB or Cryptomatte and create outline with Sobel, InPaint, Dilate-Erode kind nodes.
Option 01
For few objects you can arrange it with Index OB node and Sobel node to generate outline.


Option 02
For more complex scene Cryptomatte works better, since you can generate outlines for all objects at once from Pick output directly. You don't need to select them one by one ... just reuse random colors of Pick output.


Note: Sometimes Filter node (like Sobel, Perwitt, ... type) generates from Cryptomatte's white color filled place, Increasing Fac over 1 fix that. I would say it is because Cryptomatte's Pick colors represents more dynamic range than Filter node can handle. Using this socket directly is not proper way of using it, it is more a hack so that is why I had to use Math (Greater Than) node because direct use handled white color in a wrong way again. Greater Than node clamps high values.